Q48E Consider a solutionx1鈫抩f the l... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 1: Q48E (page 36)

Consider a solution $\overset{鈫�}{x_{1}}$ of the linear system $A \overset{鈫�}{x} = \overset{鈫�}{b}$ . Justify the facts stated in parts (a) and (b):
a. If ${\overset{鈫�}{x}}_{h}$ is a solution of the system $A \overset{鈫�}{x} = \overset{鈫�}{0}$ , then $\overset{鈫�}{x_{1}} + \overset{鈫�}{x_{h}}$ is a solution of the system $A = \overset{鈫�}{x} = \overset{鈫�}{b}$ .
b. If $\overset{鈫�}{x_{2}}$ is another solution of the system $A \overset{鈫�}{x} = \overset{鈫�}{b}$ , then $\overset{鈫�}{x_{1}} + \overset{鈫�}{x_{h}}$ is a solution of the system $A \overset{鈫�}{x} + \overset{鈫�}{0}$ .
c. Now suppose A is a $2 脳 2$ matrix. A solution vector $\overset{鈫�}{x_{1}}$ of the system $A \overset{鈫�}{x} + \overset{鈫�}{b}$ is shown in the accompanying figure. We are told that the solutions of the system $A \overset{鈫�}{x} = \overset{鈫�}{0}$ form the line shown in the sketch. Draw the line consisting of all solutions of the system $A \overset{鈫�}{x} = \overset{鈫�}{b}$ .
If you are puzzled by the generality of this problem, think about an example first:
$A = (\begin{array}{ll} 1 鈥呪赌呪赌呪赌� 2 \\ 3 鈥呪赌呪赌呪赌� 6 \end{array}), \overset{鈫�}{b} = [\begin{array}{l} 3 \\ 9 \end{array}] and \overset{鈫�}{x_{1}} = [\begin{array}{l} 1 \\ 1 \end{array}]$

Short Answer

Expert verified

a. $\overset{鈫�}{x_{1}} + \overset{鈫�}{x_{h}}$ is a solution of the system $A \overset{鈫�}{x} = \overset{鈫�}{b}$ .

b. $\overset{鈫�}{x_{2}} + \overset{鈫�}{x_{1}}$ is a solution of the system $A \overset{鈫�}{x} = \overset{鈫�}{0}$ .

c. The line consisting of all the solutions of the system $A \overset{鈫�}{x} = \overset{鈫�}{b}$ is,

Step by step solution

Consider the system.

If is an matrix with row vectors ${\overset{鈫�}{蝇}}_{1}, . . . . ., {\overset{鈫�}{蝇}}_{n}$ and $\overset{鈫�}{x}$ is a vector in ${鈩�}^{m}$ then,

$A \overset{鈫�}{x} = [\begin{matrix} 鈭� {\overset{鈫�}{蝇}}_{1} 鈭� \\ 鈰� \\ {\overset{鈫�}{蝇}}_{n} 鈭� \end{matrix}] \overset{鈫�}{x} = [\begin{matrix} 鈭� {\overset{鈫�}{蝇}}_{1} 鈰� \overset{鈫�}{x} 鈭� \\ 鈰� \\ 鈭� {\overset{鈫�}{蝇}}_{n} 鈰� \overset{鈫�}{x} 鈭� \end{matrix}]$

Consider a linear system,

$A \overset{鈫�}{x} = \overset{鈫�}{b}$

Determine the solution for the system.

${\overset{鈫�}{x}}_{1}$ is a solution of the system, $A \overset{鈫�}{x} = \overset{鈫�}{b}$ .......(1).

${\overset{鈫�}{x}}_{h}$ is a solution of the system, $A \overset{鈫�}{x} = \overset{鈫�}{0}$ .

When $\overset{鈫�}{b} = 0, A \overset{鈫�}{x} = \overset{鈫�}{b}$ ,

Thus, $\overset{鈫�}{x_{h}}$ is a solution of the system, $A \overset{鈫�}{x} = \overset{鈫�}{b} . . . . . (2)$

From the equations (1) and (2), it鈥檚 clear that,

$\overset{鈫�}{x_{1}} + \overset{鈫�}{x_{h}}$ is also a solution of the system $A \overset{鈫�}{x_{1}} + \overset{鈫�}{b}$ .

Determine the solution for the system.

${\overset{鈫�}{x}}_{1}$ is a solution of the system, $A \overset{鈫�}{x} = \overset{鈫�}{b}$ .

${\overset{鈫�}{x}}_{2}$ is a solution of the system, $A \overset{鈫�}{x} = \overset{鈫�}{b}$ .

When $\overset{鈫�}{b} = 0$ ,

$A \overset{鈫�}{x} = \overset{鈫�}{b} 鈬� A \overset{鈫�}{x} 鈫� = \overset{鈫�}{0}$

Thus, $\overset{鈫�}{x_{1}}$ is a solution of the system, $A \overset{鈫�}{x_{1}} = \overset{鈫�}{0}$ 鈥︹€� (1)

is a solution of the system, $A \overset{鈫�}{x} = \overset{鈫�}{0}$ 鈥︹赌�(2)

From the equations (1) and (2), it鈥檚 clear that,

$\overset{鈫�}{x_{2}} = {\overset{鈫�}{x}}_{1}$ is also a solution of the system $A \overset{鈫�}{x} = \overset{鈫�}{0}$ .

Consider the system

The line representing the system $A \overset{鈫�}{x} = \overset{鈫�}{0}$ must be parallel to the line representing the system $A \overset{鈫�}{x} = \overset{鈫�}{b}$ and must pass through the solution ${\overset{鈫�}{x}}_{1}$ .